Properties

Label 525.4.a.a
Level $525$
Weight $4$
Character orbit 525.a
Self dual yes
Analytic conductor $30.976$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.9760027530\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 5q^{2} + 3q^{3} + 17q^{4} - 15q^{6} - 7q^{7} - 45q^{8} + 9q^{9} + O(q^{10}) \) \( q - 5q^{2} + 3q^{3} + 17q^{4} - 15q^{6} - 7q^{7} - 45q^{8} + 9q^{9} + 12q^{11} + 51q^{12} - 30q^{13} + 35q^{14} + 89q^{16} + 134q^{17} - 45q^{18} - 92q^{19} - 21q^{21} - 60q^{22} - 112q^{23} - 135q^{24} + 150q^{26} + 27q^{27} - 119q^{28} - 58q^{29} - 224q^{31} - 85q^{32} + 36q^{33} - 670q^{34} + 153q^{36} + 146q^{37} + 460q^{38} - 90q^{39} + 18q^{41} + 105q^{42} - 340q^{43} + 204q^{44} + 560q^{46} - 208q^{47} + 267q^{48} + 49q^{49} + 402q^{51} - 510q^{52} + 754q^{53} - 135q^{54} + 315q^{56} - 276q^{57} + 290q^{58} + 380q^{59} + 718q^{61} + 1120q^{62} - 63q^{63} - 287q^{64} - 180q^{66} - 412q^{67} + 2278q^{68} - 336q^{69} - 960q^{71} - 405q^{72} - 1066q^{73} - 730q^{74} - 1564q^{76} - 84q^{77} + 450q^{78} + 896q^{79} + 81q^{81} - 90q^{82} - 436q^{83} - 357q^{84} + 1700q^{86} - 174q^{87} - 540q^{88} - 1038q^{89} + 210q^{91} - 1904q^{92} - 672q^{93} + 1040q^{94} - 255q^{96} + 702q^{97} - 245q^{98} + 108q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−5.00000 3.00000 17.0000 0 −15.0000 −7.00000 −45.0000 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 525.4.a.a 1
3.b odd 2 1 1575.4.a.l 1
5.b even 2 1 105.4.a.b 1
5.c odd 4 2 525.4.d.a 2
15.d odd 2 1 315.4.a.a 1
20.d odd 2 1 1680.4.a.u 1
35.c odd 2 1 735.4.a.j 1
105.g even 2 1 2205.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.b 1 5.b even 2 1
315.4.a.a 1 15.d odd 2 1
525.4.a.a 1 1.a even 1 1 trivial
525.4.d.a 2 5.c odd 4 2
735.4.a.j 1 35.c odd 2 1
1575.4.a.l 1 3.b odd 2 1
1680.4.a.u 1 20.d odd 2 1
2205.4.a.b 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(525))\):

\( T_{2} + 5 \)
\( T_{11} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 5 T + 8 T^{2} \)
$3$ \( 1 - 3 T \)
$5$ 1
$7$ \( 1 + 7 T \)
$11$ \( 1 - 12 T + 1331 T^{2} \)
$13$ \( 1 + 30 T + 2197 T^{2} \)
$17$ \( 1 - 134 T + 4913 T^{2} \)
$19$ \( 1 + 92 T + 6859 T^{2} \)
$23$ \( 1 + 112 T + 12167 T^{2} \)
$29$ \( 1 + 58 T + 24389 T^{2} \)
$31$ \( 1 + 224 T + 29791 T^{2} \)
$37$ \( 1 - 146 T + 50653 T^{2} \)
$41$ \( 1 - 18 T + 68921 T^{2} \)
$43$ \( 1 + 340 T + 79507 T^{2} \)
$47$ \( 1 + 208 T + 103823 T^{2} \)
$53$ \( 1 - 754 T + 148877 T^{2} \)
$59$ \( 1 - 380 T + 205379 T^{2} \)
$61$ \( 1 - 718 T + 226981 T^{2} \)
$67$ \( 1 + 412 T + 300763 T^{2} \)
$71$ \( 1 + 960 T + 357911 T^{2} \)
$73$ \( 1 + 1066 T + 389017 T^{2} \)
$79$ \( 1 - 896 T + 493039 T^{2} \)
$83$ \( 1 + 436 T + 571787 T^{2} \)
$89$ \( 1 + 1038 T + 704969 T^{2} \)
$97$ \( 1 - 702 T + 912673 T^{2} \)
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